2014/06/15 by А. N. Abyzov, Abyzov, A. N., Tran Hoai Ngoc Nhan +1
Computer Science · Mathematics · #16D80 #Advanced Algebra and Logic #FOS: Mathematics #Primary 16D10 #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #Secondary 16D40
paper · pdf · doi:10.48550/arxiv.1406.3813
openalex publication_date 2014/06/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we introduce and study the concept of CS-Rickart modules, that is a module analogue of the concept of ACS rings. A ring R is called a right weakly semihereditary ring if every its finitly generated right ideal is of the form P⊕ S, where PR is a projective module and SR is a singular module. We describe the ring R over which Matn (R) is a right ACS ring for any n ∈ \mathbb N. We show that every finitely generated projective right R-module will to be a CS-Rickart module, is precisely when R is a right weakly semihereditary ring. Also, we prove that if R is a right weakly semihereditary ring, then every finitely generated submodule of a projective right R-module has the form P1⊕ …⊕ Pn⊕ S, where every P1, …, Pn is a projective module which is isomorphic to a submodule of RR, and SR is a singular module. As corollaries we obtain some well-known properties of Rickart modules and semihereditary rings.